It is clear that the center of motion and relative motion couple.
For this form of wavefunctions, our eigenvalue problem,
\(\hat{H}_0 \Psi = E_0 \Psi \) ,
can be reduced to \(\big ( \Phi ^* \hat{H}_0 \Phi \big ) \varphi = \hat{H}_0^{\prime } \varphi = E_0 \varphi \) ,
whereWe found that our sign of the \(\vec{B}_q \cdot \hat{\vec{l}}\)  term is in conflict with Refs.[1]} and [2]}.
(reminder: \(\mu = m_1 m_2/M\) )
\(\hat{H}_0^{\prime }&=& M+ \frac{\, \vec{K} ^2 \,}{\, 2M \,}+ \frac{\, \hat{\vec{p}}^2 \,}{\, 2\mu \,} + \alpha \hat{\vec{r}}^2 - \hat{ \vec{\mu } } \cdot \vec{B}\nonumber \\&&+\, \frac{\, B_q^2 \,}{\, 8\mu \,} \hat{\vec{r}}_\perp ^2 - \frac{1}{\, M \,} \vec{B}_q \cdot ( \hat{\vec{r}} \times \vec{K} )- g_{\Delta m} \vec{B}_q \cdot \hat{\vec{l}}\,.\) 
